$A = sum_{n=0}^infty a_n$ and $b_n to B$ implies $sum_{k=0}^n a_k b_{n-k} to AB$
$begingroup$
This question is motivated by the answer With $y_n$ a sequence of real numbers, prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges,
where essentially the following fact is used:
Let $A = sum_{n=0}^infty a_n$ be an absolutely convergent series,
and $(b_n)$ a convergent sequence, $b_n to B$.
Then
$$
lim_{n to infty} sum_{k=0}^n a_k b_{n-k} = A B , .
$$
This is not too difficult to prove (sketch): Write
$$
sum_{k=0}^n a_k b_{n-k} = B sum_{k=0}^n a_k +
sum_{k=0}^n a_k bigl( b_{n-k} - B bigr)
$$
The first sum converges to $AB$. For $varepsilon > 0$, split the second sum into two parts
$$
sum_{k=0}^{n-N} a_k bigl( b_{n-k} - B bigr) + sum_{k=n-N+1}^n a_k bigl( b_{n-k} - B bigr) \
= sum_{k=0}^{n-N} a_k bigl( b_{n-k} - B bigr) + sum_{j=0}^{N-1} a_{n-j} bigl( b_j - B bigr)
$$
where $N$ is chosen such that $lvert b_n - B rvert < varepsilon$
for $n ge N$. The first part can be estimated by $varepsilon
sum_{n=0}^infty |a_n|$, and the second (finite) sum converges to zero.
Now I am fairly sure that this is not new and must have been done before.
However, I could not find a reference. So my question is:
Is there a name for the above statement, or
is there some "well-known theorem" for which this is just a special case?
The term $ sum_{k=0}^n a_k b_{n-k}$ reminds me of the Cauchy product,
but nothing is given about $sum b_n$ here. Or is it perhaps
related to summation methods for series?
(Or is it so trivial that everybody just knows it?)
sequences-and-series reference-request alternative-proof
edited Apr 13 '17 at 12:21
Community♦
1
asked Jun 12 '16 at 17:47
Martin RMartin R
28.1k33355
$endgroup$
add a comment |
$begingroup$
This question is motivated by the answer With $y_n$ a sequence of real numbers, prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges,
where essentially the following fact is used:
Let $A = sum_{n=0}^infty a_n$ be an absolutely convergent series,
and $(b_n)$ a convergent sequence, $b_n to B$.
Then
$$
lim_{n to infty} sum_{k=0}^n a_k b_{n-k} = A B , .
$$
This is not too difficult to prove (sketch): Write
$$
sum_{k=0}^n a_k b_{n-k} = B sum_{k=0}^n a_k +
sum_{k=0}^n a_k bigl( b_{n-k} - B bigr)
$$
The first sum converges to $AB$. For $varepsilon > 0$, split the second sum into two parts
$$
sum_{k=0}^{n-N} a_k bigl( b_{n-k} - B bigr) + sum_{k=n-N+1}^n a_k bigl( b_{n-k} - B bigr) \
= sum_{k=0}^{n-N} a_k bigl( b_{n-k} - B bigr) + sum_{j=0}^{N-1} a_{n-j} bigl( b_j - B bigr)
$$
where $N$ is chosen such that $lvert b_n - B rvert < varepsilon$
for $n ge N$. The first part can be estimated by $varepsilon
sum_{n=0}^infty |a_n|$, and the second (finite) sum converges to zero.
Now I am fairly sure that this is not new and must have been done before.
However, I could not find a reference. So my question is:
Is there a name for the above statement, or
is there some "well-known theorem" for which this is just a special case?
The term $ sum_{k=0}^n a_k b_{n-k}$ reminds me of the Cauchy product,
but nothing is given about $sum b_n$ here. Or is it perhaps
related to summation methods for series?
(Or is it so trivial that everybody just knows it?)
sequences-and-series reference-request alternative-proof
edited Apr 13 '17 at 12:21
Community♦
1
asked Jun 12 '16 at 17:47
Martin RMartin R
28.1k33355
$endgroup$
$begingroup$
I don't recognise the result, so it probably doesn't have a well-known name. I think the closest you get to an actual name might be something like Exercise 2.5.3 or Example 3.5 in some specified analysis or calculus textbook somewhere. It's definitely not "so trivial", and several results that are trivial in hindsight do have their own name. I just don't think this result is important enough for that.
$endgroup$
– Arthur
Jun 12 '16 at 17:52
$begingroup$
I've only ever seen this result addressed as the "reverse sum product function."
$endgroup$
– Steven Harding
Jun 12 '16 at 18:07
add a comment |
$begingroup$
This question is motivated by the answer With $y_n$ a sequence of real numbers, prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges,
where essentially the following fact is used:
Let $A = sum_{n=0}^infty a_n$ be an absolutely convergent series,
and $(b_n)$ a convergent sequence, $b_n to B$.
Then
$$
lim_{n to infty} sum_{k=0}^n a_k b_{n-k} = A B , .
$$
This is not too difficult to prove (sketch): Write
$$
sum_{k=0}^n a_k b_{n-k} = B sum_{k=0}^n a_k +
sum_{k=0}^n a_k bigl( b_{n-k} - B bigr)
$$
The first sum converges to $AB$. For $varepsilon > 0$, split the second sum into two parts
$$
sum_{k=0}^{n-N} a_k bigl( b_{n-k} - B bigr) + sum_{k=n-N+1}^n a_k bigl( b_{n-k} - B bigr) \
= sum_{k=0}^{n-N} a_k bigl( b_{n-k} - B bigr) + sum_{j=0}^{N-1} a_{n-j} bigl( b_j - B bigr)
$$
where $N$ is chosen such that $lvert b_n - B rvert < varepsilon$
for $n ge N$. The first part can be estimated by $varepsilon
sum_{n=0}^infty |a_n|$, and the second (finite) sum converges to zero.
Now I am fairly sure that this is not new and must have been done before.
However, I could not find a reference. So my question is:
Is there a name for the above statement, or
is there some "well-known theorem" for which this is just a special case?
The term $ sum_{k=0}^n a_k b_{n-k}$ reminds me of the Cauchy product,
but nothing is given about $sum b_n$ here. Or is it perhaps
related to summation methods for series?
(Or is it so trivial that everybody just knows it?)
sequences-and-series reference-request alternative-proof
edited Apr 13 '17 at 12:21
Community♦
1
asked Jun 12 '16 at 17:47
Martin RMartin R
28.1k33355
$endgroup$
This question is motivated by the answer With $y_n$ a sequence of real numbers, prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges,
where essentially the following fact is used:
Let $A = sum_{n=0}^infty a_n$ be an absolutely convergent series,
and $(b_n)$ a convergent sequence, $b_n to B$.
Then
$$
lim_{n to infty} sum_{k=0}^n a_k b_{n-k} = A B , .
$$
This is not too difficult to prove (sketch): Write
$$
sum_{k=0}^n a_k b_{n-k} = B sum_{k=0}^n a_k +
sum_{k=0}^n a_k bigl( b_{n-k} - B bigr)
$$
The first sum converges to $AB$. For $varepsilon > 0$, split the second sum into two parts
$$
sum_{k=0}^{n-N} a_k bigl( b_{n-k} - B bigr) + sum_{k=n-N+1}^n a_k bigl( b_{n-k} - B bigr) \
= sum_{k=0}^{n-N} a_k bigl( b_{n-k} - B bigr) + sum_{j=0}^{N-1} a_{n-j} bigl( b_j - B bigr)
$$
where $N$ is chosen such that $lvert b_n - B rvert < varepsilon$
for $n ge N$. The first part can be estimated by $varepsilon
sum_{n=0}^infty |a_n|$, and the second (finite) sum converges to zero.
Now I am fairly sure that this is not new and must have been done before.
However, I could not find a reference. So my question is:
Is there a name for the above statement, or
is there some "well-known theorem" for which this is just a special case?
The term $ sum_{k=0}^n a_k b_{n-k}$ reminds me of the Cauchy product,
but nothing is given about $sum b_n$ here. Or is it perhaps
related to summation methods for series?
(Or is it so trivial that everybody just knows it?)
sequences-and-series reference-request alternative-proof
sequences-and-series reference-request alternative-proof
edited Apr 13 '17 at 12:21
Community♦
1
asked Jun 12 '16 at 17:47
Martin RMartin R
28.1k33355
edited Apr 13 '17 at 12:21
Community♦
1
asked Jun 12 '16 at 17:47
Martin RMartin R
28.1k33355
edited Apr 13 '17 at 12:21
Community♦
1
edited Apr 13 '17 at 12:21
Community♦
1
edited Apr 13 '17 at 12:21
Community♦
1
1
asked Jun 12 '16 at 17:47
Martin RMartin R
28.1k33355
asked Jun 12 '16 at 17:47
Martin RMartin R
28.1k33355
asked Jun 12 '16 at 17:47
Martin RMartin R
28.1k33355
28.1k33355
$begingroup$
I don't recognise the result, so it probably doesn't have a well-known name. I think the closest you get to an actual name might be something like Exercise 2.5.3 or Example 3.5 in some specified analysis or calculus textbook somewhere. It's definitely not "so trivial", and several results that are trivial in hindsight do have their own name. I just don't think this result is important enough for that.
$endgroup$
– Arthur
Jun 12 '16 at 17:52
$begingroup$
I've only ever seen this result addressed as the "reverse sum product function."
$endgroup$
– Steven Harding
Jun 12 '16 at 18:07
add a comment |
$begingroup$
I don't recognise the result, so it probably doesn't have a well-known name. I think the closest you get to an actual name might be something like Exercise 2.5.3 or Example 3.5 in some specified analysis or calculus textbook somewhere. It's definitely not "so trivial", and several results that are trivial in hindsight do have their own name. I just don't think this result is important enough for that.
$endgroup$
– Arthur
Jun 12 '16 at 17:52
$begingroup$
I've only ever seen this result addressed as the "reverse sum product function."
$endgroup$
– Steven Harding
Jun 12 '16 at 18:07
$begingroup$
I don't recognise the result, so it probably doesn't have a well-known name. I think the closest you get to an actual name might be something like Exercise 2.5.3 or Example 3.5 in some specified analysis or calculus textbook somewhere. It's definitely not "so trivial", and several results that are trivial in hindsight do have their own name. I just don't think this result is important enough for that.
$endgroup$
– Arthur
Jun 12 '16 at 17:52
$begingroup$
I don't recognise the result, so it probably doesn't have a well-known name. I think the closest you get to an actual name might be something like Exercise 2.5.3 or Example 3.5 in some specified analysis or calculus textbook somewhere. It's definitely not "so trivial", and several results that are trivial in hindsight do have their own name. I just don't think this result is important enough for that.
$endgroup$
– Arthur
Jun 12 '16 at 17:52
$begingroup$
I've only ever seen this result addressed as the "reverse sum product function."
$endgroup$
– Steven Harding
Jun 12 '16 at 18:07
$begingroup$
I've only ever seen this result addressed as the "reverse sum product function."
$endgroup$
– Steven Harding
Jun 12 '16 at 18:07
add a comment |
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$begingroup$
I don't recognise the result, so it probably doesn't have a well-known name. I think the closest you get to an actual name might be something like Exercise 2.5.3 or Example 3.5 in some specified analysis or calculus textbook somewhere. It's definitely not "so trivial", and several results that are trivial in hindsight do have their own name. I just don't think this result is important enough for that.
$endgroup$
– Arthur
Jun 12 '16 at 17:52
$begingroup$
I've only ever seen this result addressed as the "reverse sum product function."
$endgroup$
– Steven Harding
Jun 12 '16 at 18:07